Inverse Theorems for Point-Sphere Incidences over Finite Fields

Abstract

We prove unconditional inverse theorems for point-sphere incidences over ${\mathbb{F}}_q^d$ $d\ge 2$, at the fixed-radius deviation scale. The full sphere system satisfies the centered design identity $B{B}^T=(q^d-q^{d-1})Id,$ giving the sharp mixed-radius bound $|\ddot{I}ƒ(P,S)|\le \sqrt{q^d-q^{d-1}}\sqrt{|P||S|}$ If a nonempty point set $P$ and a family of distinct spheres satisfy $\ddot{I}ƒ(P,\hat{A}\S )\ge K{q}^{(d-1)/2}\sqrt{|P||S|}$ and $\frac{64q}{K^2}\le |S|\le C_{1}K{q}^{(d-1)/2},$ then the radical hyperplane of two spheres contains at least $\frac{K{q}^{(d-1)/2}}{8C_1}$ points of $P$, and at least $\frac{3}{8}{K}^2|S|$ ordered pairs witness this. Iterating in the positive-excess core gives a two-sided decomposition into positive-surplus hyperplane pieces, together with a quantified converse. Singular pencils show sharpness in odd dimensions and, using radius-zero spheres, in split even dimensions. We also prove an incidence-rich refinement, an exact affine-hyperplane design identity, unconditional pin-count estimates, a point-hyperplane inverse theorem, and moderate-family inverse theorems for deficient pinned distances and pinned dot products.

20 pages